We report on one-dimensional photonic crystals designed to exhibit a pronounced form birefringence at terahertz frequencies. The crystals can be used as volumetric quasioptical elements for a broad frequency range. Theoretical simulations of the dielectric parameters of these structures are presented as well as measurement results of a polymeric crystal that exhibit a birefringence of 0.25 at 300 GHz. As a potential application, the device is exemplarily used as terahertz wave plate.
©2010 Optical Society of America
The terahertz (THz) frequency range is nowadays explored at an amazing pace. It is expected that THz technology will profit from birefringent materials and devices in a similar way as this is the case for optical systems. Several materials with terahertz birefringence have been reported [1–4]. Furthermore, a variety of birefringence based devices have already been demonstrated, including wave plates [5,6], filters  and phase shifters [8,9].
Recently, birefringent sub-wavelength photonic structures for optical frequencies were investigated [10–12]. These structures base on a surface modification of semiconductor materials by etching small grooves into a wafer. This results in a periodic sequence of air and semiconductor material. The birefringence associated with this structure is often called form birefringence . The resulting refractive indices can be adjusted by varying the dimensions of the grooves. Yet, the drawback of this technique is that a controlled surface etching is usually limited to depths of 50 µm to 100 µm. While these sub-100 µm thick structures can be used to build for instance Bragg filter elements for THz frequencies , the small interaction length with the THz wave counteracts a general applicability for THz birefringent devices e.g. for THz wave plates.
In this paper we advance the form birefringence approach for the THz frequency range. We present a volumetric one-dimensional photonic crystal (PC) which is formed by sheets of two different alternating polymer layers. This volume element exhibits a pronounced birefringence determined by the design parameters such as the employed material systems and the filling factor. In general, one-dimensional PCs for the THz frequency range are studied as reflectors and filter elements [14–16]. Especially designs based on alternating polymer sheets  are a promising approach due to the ease of fabrication and their low prices. Here, we propose the application of polymer based photonic crystals as customized birefringent materials.
The paper is organized as follows. First, we give a short theoretical introduction into effective medium theories which describe isotropic material composites. We show that these composite structures can be employed to construct PCs which exhibit form birefringence. Then we discuss the design issues of the proposed devices. To experimentally verify our deductions, we finally present and discuss measured data of a polymeric sample of a one-dimension PC.
To describe the interaction of a composite material system and a THz wave a quasi-static effective medium theory (EMT) can be utilized if the wavelength of the wave is larger than the dimensions of the individual components . In this case, the wave interacts with the structure in the same way as with a bulk material that exhibits effective optical parameter. Furthermore, a periodical structure of these materials results in an effective birefringence of the system .
Here, we investigate a structure as illustrated in the Fig. 1 , consisting of alternating layers of two materials with different refractive indices n1 and n2 and thicknesses L1 and L2. The resulting optical properties of this system can be described analytically in the case of a quasi-static regime analogously to a plate capacitance . The quasi-static refractive indices nTE,0 and nTM,0 for a polarization of the electrical field parallel and perpendicular to the layer, respectively, are given by:19] has analyzed this specific problem for the case of wavelengths below but still close to the quasi-static limit. He derived a 2nd order EMT which describes the effective optical material parameters nTE and nTM , as a function of the layer properties and the wavelength of the incident electromagnetic wave λ:19] which limits the frequency window of operation for the device. Thus, the dimensions of the layers have to be customized for the specific wavelength of interest. Moreover, a larger difference in the refractive indices of the utilized sheet materials will increase the resulting birefringence.
For the experimental validation of the proposed design, we fabricate a polymeric sample consisting of 250µm thick layers of polypropylene (PP) and 150 µm thick layers of a compound of PP loaded with 30% of titania (titanium dioxide). The titania particles have a mean size of less than 100nm. At frequencies around 300 GHz the refractive indices of PP and the compound are 1.5 and 2.4, respectively. The compound itself represents an effective medium. Its effective refractive index can be calculated by EMTs as discussed in . Both materials are basically non-dispersive in the frequency window investigated . The individual thicknesses were chosen for a design frequency around 300 GHz.
While coextrusion of the alternating layers is an option for the fabrication of devices of high quality without air inclusions , we build a device based on stacked polymer layers to validate the basic principle. The sheets were laser cut to a size of 5mm x 10mm. Afterwards, the alternating sheets were stacked to form a 10mm high device. To characterize the sample, we employed a conventional THz transmission time domain spectrometer as described in detail e.g. in . To extract the resulting refractive indices from the measured data the algorithm presented in  was utilized.
The resulting THz signals are shown in Fig. 2 for the case of an orientation of the sheets parallel (TE) and perpendicular (TM) to the polarization of the electrical THz field. For both orientations the THz waveform of the reference signal is chirped due to the dispersion of the device. Moreover, the THz pulse for the TE case is noticeable delayed compared to the TM case which implies different resulting optical lengths for the two polarizations of the electrical field.
The extracted refractive indices are shown in Fig. 3 compared with simulation results utilizing Eq. (2) and Eq. (3). The inset of the figure shows the refractive indices of PP and the PP compound. In the simulation we assumed resulting air layers between the polymeric sheets of approximately 50µm caused be the imperfect stacking technique . We estimate the dimension of the air spacing by a comparison of the measured device’s thickness to the sum of the thicknesses of the individual layers.
The theoretical simulations agree very well with the measured data below 400 GHz. In this frequency range, a pronounced dispersion occurs as predicted from the model. However, the resulting birefringence is relatively constant and measures 0.25. At higher frequencies, the assumptions of the model are violated since the spatial period of the structure reaches the dimension of the wavelength and scattering effects become dominant. In this frequency range, the measured data deviate from the theory’s prediction. As can been seen from the figure, the deviations are in opposite directions for the TE and TM waves and thus, the resulting birefringence decreases. This is expected since a structure with much larger dimensions than the wavelengths would not act as an effective medium and does not exhibit any form birefringence.
To demonstrate one possible application, we adjusted the crystal under an angle of 45° with respect to the polarization of the THz wave. In this configuration the crystal acts as a wave plate and rotates the polarization of the wave as a function of the frequency. This can be characterized by analyzing the resulting transmittance. The wave plate rotates the polarization of the electrical field by 90° if the relative phase shift between the two field components Δφ=2 πLΔn/λ, where Δn is the birefringence and L the device thickness, reaches a multiple of π. In the case of Δφ= π/2 3π/2,… the devices acts as quarter wave plate .
The amplitude transmittance T through a polarizer can be described by the Malus’ law, which is given by:
For an orientation of the polarizer parallel to the incident electrical field of the THz wave, the transmission reaches its maxima. In this case the wave plate rotates the polarization by 180° or 360°, meaning Δφ=2π, 4π,... . Minima occur in the case of a rotation by 90° or 270° that corresponds to Δφ=π, 3π,... . Thus, by considering the transmittance TTM and TTE for the individual field components of the THz wave, me model the resulting frequency depended transmission trough the polarizer by:
For the measurement a wire grid polarization filter was placed in front of the receiver antenna to ensure that only the horizontally polarized electrical field component is detected. Figure 4 shows the measured amplitude transmittance and the theoretical predictions based on Eq. (5). Two distinctive minima around 150 GHz and 375 GHz can be observed. The maximum transmission results at 250 GHz. These results agree with the calculations of the relative phase shift considering the frequency depended amount of birefringence. For frequencies above 380 GHz, the crystal does no longer act as a perfect effective medium since the wavelength reaches the size of the layers. Thus, the transmittance of the wave plate comprises deviations from the theoretical shape e.g. the features around 400 GHz. The reason for that is that the effective birefringence reduces in this frequency range as can be seen in Fig. 3. Nevertheless, the modeled transmission, which is based on Eq. (5), the measured refractive indices and the individual transmittance for the TE and TM waves, is in good agreement to the measured data.
Due to the absorption of the titania loaded PP the maximum transmittance measures 65% only. However, polymers loaded with additives that exhibit lower absorption coefficients, like sapphire particles, or a combination of polymer and air layers could increase the transparency of future devices.
We presented an approach for realizing polymer based one-dimensional photonic crystals with pronounced birefringence. By varying the materials employed and the layer thicknesses, the resulting refractive indices and therefore the resulting birefringence can be adjusted. Potential applications include THz wave plates and frequency filter elements. The resulting dispersion could also be utilized for pulse shaping purposes. While coextruded polymeric stacks could provide even higher efficiencies, the first proof of principle device already illuminates the possibilities of this design approach.
We thank the Süddeutsches Kunststoff-Zentrum, Würzburg, as well as Steffen Wietzke for providing the PP compound.
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